Analysis reveals asymptotic periodicity in multidimensional contracted rotations for almost all b, indicating significant implications for ergodic theory.
We study the dynamics of multidimensional contracted rotations and address a problem posed by Y. Bugeaud and J-P. Conze in Acta Arithmetica in 1999. More precisely, we show that if A is an invertible linear contraction of Rᵈ, then the map f: [0,1)ᵈ→ [0,1)ᵈ defined by f(x) = Ax +b\,\,(mod\,Zᵈ) is asymptotically periodic for Lebesgue almost all bᵈ.
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Gaivão et al. (2025) studied this question.
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